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My brain seems to have shut down. Lets say I'm working in the space [itex] H_a \otimes H_b [/itex] and I have an operator A and B on each of these spaces respectively. Furthermore, consider a general state in the tensor space [itex] |\psi \rangle = | x_1 \rangle |y_1\rangle + |x_2 \rangle |y_2 \rangle [/itex] (ignoring normalization). Now in my head, I keep on thinking that if I wanted to apply the operator [itex] A \otimes B [/itex] to [itex] | \psi \rangle [/itex] I couldn't simply go

[tex] A \otimes B | \psi \rangle = A| x_1 \rangle B|y_1\rangle + A|x_2 \rangle B|y_2 \rangle [/tex]

However, I was playing around with some values today (assuming finite dimensions), and from what I tried it always seemed to work. Can we do this in general, or did I just get lucky in all my trials?

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# Operators on Tensor States

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